A few possible answers:
From a more secular view:
Heinrich Zimmern in The Babylonian and the Hebrew Genesis (Originally in German "Biblische und babylonische Urgeschichte"), pg 44-45, wrote:
"...We have found in Babylonian traditions the origin of this figure of the sage walking with God through the greater part of a life of three hundred and sixty-five years. Of the seventh primaeval king - Evedoranchos, as the name is given in Greek form by Berossus, or Enmeduranki, as it is given in Babylonian - many tales are told. He was king of Sippar, the city of the Sun-god Shamash. The god called him to intercourse with himself, taught him concerning many secrets of heaven and earth, and especially the art of foretelling the future from signs in heaven and on earth. Thus Enmeduranki became to the Babylonians the prototype of soothsayers...and, above all, by astrology, that is, the reading of the future by observation of the movements of the heavenly bodies and the positions of the constellations. As Enmeduranki to the Babylonians so did Henoch stand to the Jews of later times as the father and founder of astrology, and, indeed, of all knowledge in heaven and on earth. His life of three hundred and sixty-five years - short compared with the great ages attributed to the other patriarchs - may now be explained easily: Enmeduranki being in the service of the Sun-god, to Henoch were attributed the number of the days of the solar year as the number of the years of his life.
Therefore, the significance of the number of years of Enoch's life being the same number of days as the solar year was to emphasize that Enoch was the father of astrology - and possibly other paths of wisdom1 - while removing the polytheistic aspect of that tradition, which was, of course, contradictory to the Israelite religion.
From a more Jewish view, I can suggest:
a. Perhaps there is no sun-related symbolic significance to Enoch's age, and the Biblical author actually thought that Enoch died at age 365 and wished to convey this to the Israelites.
b. Jewish tradition holds that the Jewish calendar, was and is a solar-lunar calendar2, which, in this context, means that neither the sun nor the moon "rule" exclusively over anything. Neither is more important; rather, the one God rules over all. This would explain why the myth of Enoch ruling over Sippar was not included but his age was. Again, a removal of the polytheistic aspect of the tale.3
c. I have just stumbled upon what Philo wrote on Enoch's lifetime:
In the first place, the year contains three hundred and sixty-five days; therefore, by the symbol of the solar orbit, the sacred historian here indicates the life of the repentant man. In the second place, as the sun is the cause of day and night, performing his revolutions by day above the hemisphere of the earth, and his course by night under the earth, so also the life of the man of repentance consists of alternations of light and darkness; of darkness, that is, of times of agitation and circumstances of injury; and of light, when the light of virtue and its radiant brilliancy arises. In the third place, he has assigned to him a complete number, as the sun is ordained to be the chief of the stars of heaven, under an appointed number, in the time which came before the period of his repentance, to lead to the oblivion of the sins previously committed; since, as God is good, he bestows the greatest favours most abundantly, and, at the same time, he effaces the former offences of those who devote themselves to him, and which might deserve chastisement, by a recollection of their virtues. (The Midrash of Philo, 5:23)
Philo emphasizes an idea that was also presented by some Jewish scholars, that Enoch was a repentant man - sometimes he sinned and sometimes he repented and was righteous.4 Every year, according to the Jewish calendar, eventually brings one back to the Days of Atonement, also known as the High Holidays, during the months of Elul and Tishrei. Regardless of what that person may have done, every year, he has another chance to atone for his sins. The second point he makes ties into the first. The third also, but it may be argued that the third puts a special emphasis on the importance of the sun itself, which may be used as an argument in favor of similarities with the Mesopotamian Enoch being ruler of the sun-god-city. However, Philo emphasizes that the sun is only another tool for bringing about the glory of God.
d. Some Jewish scholars have written that the significance of the number of years matching the solar calendar is to tell us that Enoch during his lifetime came to know incredible astronomical information (and not astrological, as Zimmern put it), which in turn led him to the realization that there is a higher power, that "all the bodies of light come from the Higher Light", as Rabbienu Bechayey ibn Chalawa put it, in his commentary on the verse, or as Rabbi Yitzchak Karo put in his book Toldot Yitzchak, "...and for this he said "and Enoch walked with God", and the phrasing "walking" is only mentioned in relation to righteous ones. And by Noah it is said "Noah walked with God" and by Abraham it is said "Walk in My ways" and they all gained knowledge of the cycle of the sun and its great powers that teach about the rulership of The Lord of All and His Greatness5 and Enoch stuck to The Higher Light from Whom came and were created the bodies of light on the fourth day." (my translation)
Either way, it seems that the point made by the Biblical author was to emphasize the monotheistic aspect of the figure and the removal of the polytheistic aspect.
1 Though I am not sure based on what Jewish sources Zimmern says this.
2 Prof. Sid Z. Leiman in this lecture, I believe, makes mention of the specific Dead Sea cult that used the specific calendar you mentioned. Yes, it was inherently different from the Pharisee calendar which was later accepted by the vast majority of the Jewish people, and not just because it was a solar calendar. The tradition I refer to is the one now commonly accepted by Orthodox Jews based on the Pharisee tradition.
3 Something similar was suggested about the figures of Tubal-Cain, Lemech and Naamah by the Jewish scholars Malbim, Shadal and Rabbi Immanuel Frances, as can be seen in the translations I brought in this answer. Both ideas - about Lemech and his descendants being the basis of later myths and Enoch and his descendants being thought of as the mythical rulers of the ancient Mesopotamian cities - are also brought by Rabbi Philip Biberfeld in his book Universal Jewish History, vol. I.
4 Midrash Beresheet Rabbah 25:1: "Said Rabbi Aibo: Enoch was a hypocrite - sometimes he was righteous, sometimes he was a sinner. Said The Holy One, Blessed Is He: Until he is in his righteousness, I shall remove him [from this world]" (my translation); New Zohar, Genesis 842: "It is said in a baraita: Enoch son of Yared was a righteous man, but accepted upon himself to do evil. And The Holy One, Blessed Is He took him before his time, to do charity with him and to give him a good reward in the world to come." (my translation).
5 Rabbi Karo is referring to a well-known midrash which states that Abraham, who, after all, came from a family of idolaters, came to know of the one God by examining nature and the astronomical planetary bodies. (Midrash Hagadol, Lech Lecha 12).